The Master Density Equation for Particulate Processes

We now demonstrate how the equation for the master density is derived for the processes considered in Chapter 3. We shall select only one of them, 

2 can be obtained. In order to derive the growth term in the master density equation we adopt an approach that is only apparently different from that used in Chapter 2 for the derivation of the population balance equation. 

1 See Ramkrishna (1981). See Fox and Fan (1988) for a stochastic treatment of breakage and coalescence using the master equation approach. 

Further, as at most one breakup event may occur during the time interval dt with probability of order 0 dt at most two of the v particles at time t may be freshly formed fragments. This pair of fragments, which could be any two of the V particles present at time t, must necessarily arise from the breakage of a particle whose mass at time t — dt must equal the sum of the masses of the pair selected. Thus, 

There are several clarifications to be made in regard to the last expression on the right-hand side of the foregoing equation. First, when examining the terms in the sum the particle size with subscripts 0 and v + 1 must be ignored, since they do not exist. Second, the summation on j and k is such as to automatically avoid double counting of particle pairs. Third, the notation i = 1,7, k in the product range is used to exclude i = j as well as 

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Second, Section 7.2 provides the derivation of equations in the master density for some of the particulate processes discussed in Chapter 3 and shows the combinatorial complexity of their solution. The objective of this section is to show how Monte Carlo simulations of a process eliminate the quantitatively less significant combinatorial elements of the solution by artificial realization. Thus, the discussion here is of largely conceptual value. 

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